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A note on loops of square-free order

Emma Leppälä; Markku Niemenmaa — 2013

Commentationes Mathematicae Universitatis Carolinae

Let $Q$ be a loop such that $| Q |$ is square-free and the inner mapping group $I (Q)$ is nilpotent. We show that $Q$ is centrally nilpotent of class at most two.

On finite commutative loops which are centrally nilpotent

Emma Leppälä; Markku Niemenmaa — 2015

Commentationes Mathematicae Universitatis Carolinae

Let $Q$ be a finite commutative loop and let the inner mapping group $I (Q) ≅ C_{p^{n}} \times C_{p^{n}}$ , where $p$ is an odd prime number and $n \geq 1$ . We show that $Q$ is centrally nilpotent of class two.

On dicyclic groups as inner mapping groups of finite loops

Emma Leppälä; Markku Niemenmaa — 2016

Commentationes Mathematicae Universitatis Carolinae

Let $G$ be a finite group with a dicyclic subgroup $H$ . We show that if there exist $H$ -connected transversals in $G$ , then $G$ is a solvable group. We apply this result to loop theory and show that if the inner mapping group $I (Q)$ of a finite loop $Q$ is dicyclic, then $Q$ is a solvable loop. We also discuss a more general solvability criterion in the case where $I (Q)$ is a certain type of a direct product.

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A note on loops of square-free order

On finite commutative loops which are centrally nilpotent

On dicyclic groups as inner mapping groups of finite loops